Solid State Chemistry and its Applications. Anthony R. West

anion vacancies may be present, within the overall cp array. These vacancies are usually at random throughout the anion array, such as in the perovskite (La_1−x Sr _x )(Ga_1−x Mg _x )O_3−x , but the ordering of oxygen vacancies may occur as in, for example, the beta alumina family of structures in which every fifth layer in a cp oxide sequence has ¾ of the oxygens missing, Fig. 8.23. In other cp structures, particularly those containing large cations such as K⁺, Sr²⁺ or La³⁺ in perovskites, mixed cp layers are present because these cations are large enough to be surrounded by 12 anion neighbours. Thus, in SrTiO₃, Sr and O together form a ccp sequence in which the Sr and O positions are ordered rather than randomised, Fig. 1.41(e). And in some other structures, especially fluorite, the cations rather than the anions form a cp array. Finally, the structures may be eutactic in that the cp arrangement of one type of ion is clearly present, but the ions are pushed apart and therefore, the structures are not closest packed.

Within a cp anion array, interstitial sites for cations are restricted to either tetrahedral or octahedral, as discussed next. For the anion coordination number, however, there is no such limitation. Although the anion–anion coordination number (above) is twelve, this refers to the second coordination sphere around a particular anion whereas the anion–cation coordination number refers to the primary coordination sphere, with the smallest interatomic distances. Various examples of anion–cation coordination numbers and arrangements are given throughout this chapter, as well as examples of structures that are distorted away from an ideal cp arrangement. First, we consider the cases of the standard tetrahedral and octahedral coordinations.

       1.15.3.1 Tetrahedral and octahedral sites

Two types of interstitial site, tetrahedral and octahedral, are present in cp structures, Fig. 1.23. These may be seen by considering the nature of the interstitial space between any pair of adjacent cp layers. For the tetrahedral sites, three anions that form the base of the tetrahedron belong to one cp layer with the apex of the tetrahedron either in the layer above (a) or below (b). This gives two orientations of tetrahedral sites, T₊ and T_–, in which the apex is up and down, respectively, and is illustrated further in (c) and (d) with the layers seen edge‐on. Because the centre of gravity of a tetrahedron is nearer to the base than the apex (see Appendix C), cations in tetrahedral sites are not located midway between adjacent anion layers but are nearer one layer than the other. In ccp structures, there are four equivalent orientations of cp layers and therefore, four equivalent orientations of T₊, T_– sites

       Figure 1.23 Tetrahedral and octahedral sites between two cp anion layers, seen from different perspectives. (a, b) Projection down threefold axis of T+, T– sites. (c, d) Tetrahedral sites edge‐on. (e) Projection down threefold axis of octahedral site and (f) seen edge‐on. (g, h) Conventional representation of octahedral site. (i) Distribution of T+, T–, O sites between two cp layers. The labelling of tetrahedral sites as T+ and T− is arbitrary and is only to indicate that there are two possible different orientations for these sites.

      Octahedral sites, O, are coordinated to three anions in each layer, Fig. 1.23(e) and are midway between the anion layers (f). A more common way to regard octahedral coordination is as four coplanar atoms with one atom at each apex above and below the plane. In (e), atoms 1, 2, 4 and 6 are coplanar; 3 and 5 form apices of the octahedron. Also, atoms 2, 3, 4, 5 and 1, 3, 5, 6 are coplanar. This is further illustrated in Fig. 1.23(g) and (h), which are similar to (e) but seen from a different and more conventional perspective.

      The distribution of interstitial sites between any two adjacent layers of cp anions is shown in Fig. 1.23(i). We can see that below every red sphere in the upper layer is a T₊ site; likewise above every blue sphere in the lower layer is a T_– site. There are also as many O sites as either T₊ or T_– sites. A similar distribution to that in (i) is seen between each pair of cp anion layers. Counting up the numbers of each, then, for every anion there is one octahedral site and two tetrahedral sites, one T₊ and one T_–.

It is rare that all the interstitial sites in a cp structure are occupied; often one set is full or partly occupied and the remaining sets are empty. A selection of cp ionic structures, classified according to the anion layer stacking sequence and the occupancy of the interstitial sites, is given in Table 1.4. Individual structures are described later. Here we simply note how a wide range of structures are grouped into one large family; this helps to bring out similarities and differences between them. For example:

      1 The rock salt and nickel arsenide structures both have octahedrally coordinated cations and differ only in the anion stacking sequence. Similarly, there are other pairs of structures with similar cation coordination numbers that differ only in the anion stacking sequence, e.g. olivine and spinel, wurtzite and zinc blende, CdI2 and CdCl2.Table 1.4 Some close packed structuresAnion arrangementInterstitial sitesExamplesT+T–Occp––1NaCl, rock salt1––ZnS, blende, or sphalerite1/81/81/2MgAl2O4, spinel––1/2CdCl2––1/3CrCl311–K2O, antifluoritehcp––1NiAs1––ZnS, wurtzite––1/2CdI2––1/2TiO2, rutile^a––2/3α‐Al2O3, corundum1/81/81/2Mg2SiO4, olivineccp ‘BaO3’ layers––1/4BaTiO3, perovskite^a The hcp oxide layers in rutile are not planar but are buckled; the oxide arrangement may alternatively be described as tetragonal packed, tp.

      2 Rutile, TiO2, and CdI2 both have hcp anions (although the layers are buckled in rutile) with half the octahedral sites occupied by cations but they differ in the manner of occupancy of these octahedral sites. In rutile, half the octahedral sites between any pair of cp anion layers are occupied by Ti. In CdI2, layers of fully occupied octahedral sites alternate with layers in which all sites are empty; this gives CdI2 a layered structure.

      3 In a few structures, it is useful to regard the cation as forming the cp layers with the anions occupying the interstitial sites. The fluorite structure, CaF2, may be regarded as ccp Ca with all T+ and T– sites occupied by F. The antifluorite structure of K2O is the exact inverse of fluorite with ccp layers of O and K in T+, T– sites; it is included in Table 1.4.

      4 The cp concept may be extended to structures in which a mixture of anions and large cations form the packing layers and smaller cations occupy interstitial sites. In perovskite, BaTiO3, ccp layers of composition ‘BaO3’ occur and one‐quarter of the octahedral sites between these layers are occupied by Ti, although only those sites in which all six corners are O2– ions.

      5 Some structures are anion‐deficient cp structures in which the anions form an incomplete cp array. The ReO3 structure has ccp oxide layers with one‐quarter of the O sites empty. It is analogous to perovskite just described in which Ti is replaced by Re and Ba sites are left vacant. The structure of β‐alumina, nominally of formula NaAl11O17, contains cp oxide layers in which every fifth layer has approximately three‐quarters of the O2– ions missing.

       1.15.3.2 Relative sizes of tetrahedral and octahedral sites

      A general guideline is that tetrahedra are smaller than octahedra which, in turn, are smaller than polyhedra of higher coordination number; for a given anion array, the relative sizes of sites may be obtained quantitatively by geometric calculations (see the next section). In ionic structures, cations M occupy those sites or polyhedra